# Block design

Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

## Overview

In combinatorial mathematics, a block design (more fully, a balanced incomplete block design) is a particular kind of set system, which has long-standing applications to experimental design (an area of statistics) as well as purely combinatorial aspects.

Given a finite set X (of elements called points) and integers k, r, λ ≥ 1, we define a 2-design B to be a set of k-element subsets of X, called blocks, such that the number r of blocks containing x in X is independent of x, and the number λ of blocks containing given distinct points x and y in X is also independent of the choices.

Here v (the number of elements of X, called points), b (the number of blocks), k, r, and λ are the parameters of the design. (Also, B may not consist of all k-element subsets of X; that is the meaning of incomplete.) The design is called a (v, k, λ)-design or a (v, b, r, k, λ)-design. The parameters are not all independent; v, k, and λ determine b and r, and not all combinations of v, k, and λ are possible. The two basic equations connecting these parameters are

${\displaystyle bk=vr,}$
${\displaystyle \lambda (v-1)=r(k-1).}$

A fundamental theorem (Fisher's inequality) is that bv in any block design. The case of equality is called a symmetric design; it has many special features.

Examples of block designs include the lines in finite projective planes (where X is the set of points of the plane and λ = 1), and Steiner triple systems (k = 3). The former is a relatively simple example of a symmetric design.

## Generalization: t-designs

Given any integer t ≥ 2, a t-design B is a class of k-element subsets of X (the set of points) , called blocks, such that the number r of blocks that contain any point x in X is independent of x, and the number λ of blocks that contain any given t-element subset T is independent of the choice of T. The numbers v (the number of elements of X), b (the number of blocks), k, r, λ, and t are the parameters of the design. The design may be called a t-(v,k,λ)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosen arbitrarily. The equations are

${\displaystyle b_{i}=\lambda {\binom {v-i}{t-i}}/{\binom {k-i}{t-i}}{\text{ for }}i=0,1,\ldots ,t,}$

where bi is the number of blocks that contain any i-element set of points.

Examples include the d-dimensional subspaces of a finite projective geometry (where t = d + 1 and λ = 1).

The term block design by itself usually means a 2-design.

## References

• van Lint, J.H., and R.M. Wilson (1992), A Course in Combinatorics. Cambridge, Eng.: Cambridge University Press.
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